Acta Applicandae Mathematica

, Volume 67, Issue 2, pp 173–224 | Cite as

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

  • G. Hörmann
  • Maarten V. de Hoop

Abstract

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.

microlocal analysis hyperbolic partial differential equation generalized functions 

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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • G. Hörmann
    • 1
  • Maarten V. de Hoop
    • 1
  1. 1.Center for Wave PhenomenaColorado School of MinesGoldenUSA

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